reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];

theorem
  i <= k & j <= l implies [:Segm i,Segm j:] c= [:Segm k,Segm l:]
  proof
    assume that
A1: i <= k and
A2: j <= l;
    now
      let x be object;
      assume x in [:Segm i,Segm j:];
      then consider xi,xj be object such that
A3:   xi in Segm i and
A4:   xj in Segm j and
A5:   x = [xi,xj] by ZFMISC_1:def 2;
      reconsider xi,xj as Nat by A3,A4;
      xi < k & xj < l by A1,A2,A3,A4,NAT_1:44,XXREAL_0:2;
      then xi in Segm k & xj in Segm l by NAT_1:44;
      hence x in [:Segm k,Segm l:] by A5,ZFMISC_1:def 2;
    end;
    hence thesis;
  end;
